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NASA NTRS1976

Feasibility of a second-order gravitational red-shift experiment

The number of gravitation experiments undertaken since the advent of Einstein's theory of gravitation is quite small, with, so far, only the famous perihelion-advance experiment and a recent lunar-laser-ranging experiment being capable of measuring a nonlinear, second-order effect. It now appears that another distinct test of the second-order term may be feasible through the use of very stable atomic clocks. This experiment, which would measure the second-order gravitational red-shift, is a bona fide test of the field equations of gravity, not just a test of the underlying principle of equivalence. The nature of such an experiment, the basic equations, model-orbit calculations, and some tracking-accuracy requirements are presented. It is concluded that current space-probe tracking capabilities cannot determine all the necessary orbital parameters with sufficient accuracy for this experiment at the present time.

Jaffe, J.↗

NASA NTRS1972

Navigation strategy and filter design for solar electric missions

Methods which have been proposed to improve the navigation accuracy for the low-thrust space vehicle include modifications to the standard Sequential- and Batch-type orbit determination procedures and the use of inertial measuring units (IMU) which measures directly the acceleration applied to the vehicle. The navigation accuracy obtained using one of the more promising modifications to the orbit determination procedures is compared with a combined IMU-Standard. The unknown accelerations are approximated as both first-order and second-order Gauss-Markov processes. The comparison is based on numerical results obtained in a study of the navigation requirements of a numerically simulated 152-day low-thrust mission to the asteroid Eros. The results obtained in the simulation indicate that the DMC algorithm will yield a significant improvement over the navigation accuracies achieved with previous estimation algorithms. In addition, the DMC algorithms will yield better navigation accuracies than the IMU-Standard Orbit Determination algorithm, except for extremely precise IMU measurements, i.e., gyroplatform alignment .01 deg and accelerometer signal-to-noise ratio .07. Unless these accuracies are achieved, the IMU navigation accuracies are generally unacceptable.

Tapley, B. D.↗

NASA NTRS1974

Design of a Torque Current Generator for Strapdown Gyroscopes

The design, analysis, and experimental evaluation of an optimum performance torque current generator for use with strapdown gyroscopes, is presented. Among the criteria used to evaluate the design were the following: (1) steady-state accuracy; (2) margins of stability against self-oscillation; (3) temperature variations; (4) aging; (5) static errors drift errors, and transient errors, (6) classical frequency and time domain characteristics; and (7) the equivalent noise at the input of the comparater operational amplifier. The DC feedback loop of the torque current generator was approximated as a second-order system. Stability calculations for gain margins are discussed. Circuit diagrams are shown and block diagrams showing the implementation of the torque current generator are discussed.

Mcknight, R. D.↗

NASA NTRS1973

Classical eighth- and lower-order Runge-Kutta-Nystroem formulas with a new stepsize control procedure for special second-order differential equations

New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.

Fehlberg, E.↗

NASA NTRS1979

Capturing and tracking performance of the horizontal guidance and control systems of the terminal configured vehicle

A twin-jet commercial transport equipped with digital navigation, guidance, and control systems and advanced electronic display system was used for airborne operational research. The results of flight tests which evaluated a second-order horizontal-path guidance control law and the autopilot and airplane response are presented. This evaluation was accomplished through analysis of recorded flight data and through pilot opinion of the airplane maneuvers during automatic path-capture scenarios and path tracking. Four different path captures were flown at a ground speed of approximately 160 knots and repeated at approximately 300 knots. Path tracking tracking error was measured in terms of cross track error along two paths: one pat at cruise speeds and one at airport-terminal-area speeds. The path tracking accuracy and the smoothness of the airplane maneuvers were judged satisfactory for high speeds; however, at lower speeds the control law design should be improved so that tracking will be more accurate for operations in the airport terminal area.

Knox, C. E.↗

NASA NTRS1973

Analysis of axisymmetrical vibration of a partially liquid-filled elastic sphere by the method of Green's function

The longitudinal dynamic behavior of launch vehicles is largely determined by their huge liquid propellant masses which are spring supported by the elastic tank walls. Throughout much of the powered flight time, these masses constitute a high percentage of the entire vehicle mass and, therefore, may dominate the fundamental modes of the vehicle. In this report, a spherical container is considered. The analysis is based on a Galerkin approach, in the course of which a second-order differential equation must be solved. The solution has been obtained by the method of Green's function. This method is favorable because it displays the manner in which the analysis can be extended to partially liquid-filled general shells of revolution. The computer programs currently available for partially liquid-filled propellant tanks are based on the finite element methods and result in analytical models having as many as several hundred degrees of freedom. The method applied in this report results in a model having less than 10 degrees of freedom as can be shown by numerical evaluation. Therefore, it will be possible to analyze propellant tanks using much less computer time with comparable accuracy.

Glaser, R. F.↗

NASA NTRS1982

Vortex simulation of an inviscid shear layer

The accuracy of the vortex-blob method was tested by simulating a free-shear-layer instability, Kirchhoff's elliptical vortex, and a circular vortex. The main numerical parameters in the vortex-blob method are the density of the vortices, and the distribution of vorticity within each vortex core. The growth rate of a periodic unstable mode of the shear layer was calculated numerically and compared with the exact result. The error is only a few percent for about 10 rows of vortex blobs. The error is reduced by decreasing the spacing between vortices and, correspondingly, the core size. In the simulation of the motion of the elliptical vortex, the rotation of the boundary, without change of shape, and the circular particle paths of the vortical fluid were well simulated. For the circular vortex, optimum sets of parameters were obtained by comparing them with the exact velocity. The results are consistent with convergence theories of the vortex-blob method. In particular, second-order convergence is observed with a Gaussian core from velocity calculation.

Nakamura, Y.↗

NASA NTRS1965

Human Tracking Performance in Uncoupled and Coupled Two-Axis Systems

This report presents tile results of an experimental and analytical study of human performance in uncoupled and coupled control systems. Human pilot performance in single and two-axis systems was mathematically modeled by linear second-order describing functions. Model parameters were determined using model matching techniques. Analysis of the models showed that the amplitude ratio and phase lead of the describing function increased with training indicating an increase in open loop bandwidth. The phase margin also decreased with training. Increasing the plant lag time constant resulted in an increase in the model lead time constant and a decrease in the zero frequency gain. No significant difference was found to exist in the normalized tracking error per axis between the two-axis tasks and the single-axis tasks. However tile model lead time constant was significantly greater in two-axis tracking. Manual tracking of two-axis systems with cross-coupling was studied experimentally and analytically. Approximate methods for modeling two-axis performance were developed and checked using a precise spectral analysis approach. Coupled and uncoupled, symmetrical and asymmetrical two-axis performance was compared. The results show that modeling of cross-coupled systems is feasible and that trained subjects are capable of decoupling the axes of some systems. A methodology study compared the identification performance of continuous, iterative, and extrapolation model matching techniques. An iterative technique employing sensitivity equations for the generation of influence coefficients was found to be the best technique due to its excellent identification accuracy and ease of implementation. Convergence in iterative techniques can be improved substantially by equalizing the parameter adjustment rates and limiting the maximum correction per iteration.

Todosiev, E. P.↗

NASA NTRS1974

Classical seventh-, sixth-, and fifth-order Runge-Kutta-Nystrom formulas with stepsize control for general second-order differential equations

Runge-Kutta-Nystrom formulas of the seventh, sixth, and fifth order were derived for the general second order (vector) differential equation written as the second derivative of x = f(t, x, the first derivative of x). The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the local truncation error in x, and they require no more evaluations per step than the earlier Runge-Kutta formulas for the first derivative of x = f(t, x). The developed formulas are expected to be time saving in comparison to the Runge-Kutta formulas for first-order differential equations, since it is not necessary to convert the second-order differential equations into twice as many first-order differential equations. The examples shown saved from 25 percent to 60 percent more computer time than the earlier formulas for first-order differential equations, and are comparable in accuracy.

Fehlberg, E.↗

NASA NTRS1974

On the accuracy of Whitham's method

The steady flow of an ideal gas past a conical body is studied by the method of matched asymptotic expansions and by Whitham's method in order to assess the accuracy of the latter. It is found that while Whitham's method does not yield a correct asymptotic representation of the perturbation field to second order in regions where the flow ahead of the Mach cone of the apex is disturbed, it does correctly predict the changes of the second-order perturbation quantities across a shock (the first-order shock strength). The results of the analysis are illustrated by a special case of a flat, rectangular plate at incidence.

Zahalak, G. I.↗

NASA NTRS1974

A comparison of numerical solutions to the inviscid equations of fluid motion

The second-order MacCormack method and the third-order Rusanov and Kutler-Warming-Lomax methods are applied to the inviscid 1-d Burgers' equation, wedge flow, and the problem of shock reflection from a rigid boundary. The numerical solution in each case is compared to the exact solution and the quantitative estimates of accuracy are obtained. Results of this study show that the third-order Kutler-Warming-Lomax or the Rusanov methods tuned for minimum dissipation or minimum dispersion provide the most accurate solution in each of the examples considered.

Anderson, D. A.↗

NASA NTRS1975

Second-order non-iterative ADI solution of non-linear partial differential equations

A new method for the solution of non-linear partial differential equations by an ADI procedure is described. Although the method is second order accurate in time, it does not require either iterations or predictor corrector methods to overcome the nonlinearity of the equations. Thus the computational effort required for the solution of the non-linear problem becomes similar to that required for the linear case. The method is applied to a two-dimensional 'extended Burgers equation'. Linear stability is studied, and some numerical solutions obtained. The improved accuracy obtained by the 2nd order truncation error is clearly manifested.

Wolfshtein, M.↗

NASA NTRS1976

Multidimensional difference schemes with fourth-order accuracy

An explicit finite-difference algorithm is presented for the solution of quasilinear divergence free multidimensional hyperbolic systems. The method consists of four steps per time level. The resulting scheme is fourth-order accurate in both space and time, though the intermediate steps are only first-order accurate. The family of schemes introduced is dissipative, and hence, suitable for both smooth flows and flows containing shocks. This method is compared, in several numerical examples, with both second-order schemes and others that are fourth order in space, but second order in time.

Turkel, E.↗

NASA NTRS1977

A critical study of higher-order numerical methods for solving the boundary-layer equations

A fourth-order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations. The efficiency of the present method is compared with other two-point and three-point higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, and the three-point spline methods. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.

Wornom, S. F.↗

NASA NTRS1977

Principles of equivalence, Eotvos experiments, and gravitational red-shift experiments - The free fall of electromagnetic systems to post-post-Coulombian order

Free fall in an external static gravitational field of a composite test body consisting of electromagnetically interacting charged particles is analyzed using the TH-epsilon-mu formalism developed by Lightman and Lee (1973). The Lightman and Lee calculation that was carried out to 'post-Coulombian' order is extended to 'post-post-Coulombian' order, and violations of the weak equivalence principle are found which result from the magnetostatic self-energy of the test body. The current generation of Eotvos experiments is shown to test the cited formalism at first order in the external gravitational potential to an accuracy of 4 by 10 to the -10th power for electrostatic energies and 6 by 10 to the -6th power for magnetostatic energies. It is concluded that although the formalism can be tested to an accuracy of 0.04 for electrostatic energies at second order in the gravitational potential, a significant second-order test for magnetostatic energies would require Eotvos experiments accurate to better than one part in 10 to the 15th power.

Haugan, M. P.↗

NASA NTRS1978

Application of higher-order numerical methods to the boundary-layer equations

A fourth-order method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations for both attached and separated flows. The efficiency of the present method is compared with other higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, the three-point spline methods, and a modified finite-element method. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.

Wornom, S. F.↗

NASA NTRS1979

Second-order estimates in singular perturbation of output regulator problems

A method of order reduction in singular perturbation problems due to Chow and Kokotovic is studied. The method is used for an output regulator problem in control theory. The quadratic term in the asymptotic expansion of Chow and Kokotovic is calculated and it is shown by some examples how this term can give evidence of the accuracy of the procedure.

Rublein, G.↗

NASA NTRS1976

Polynominal Interpolation Methods for Viscous Flow Calculations

Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and by Hermitian (Taylor series) finite-difference discretization. The similarities and special features of these different developments are discussed. The governing systems apply for both uniform and variable meshes. Hybrid schemes resulting from two different polynomial approximations for the first and second derivatives lead to a nonuniform mesh extension of the so-called compact or Pad? difference technique (Hermite 4). A variety of fourth-order methods are described and the Hermitian approach is extended to sixth-order (Hermite 6). The appropriate spline boundary conditions are derived for all procedures. For central finite differences, this leads to a two-point, second-order accurate generalization of the commonly used three-point end-difference formula. Solutions with several spline and Hermite procedures are presented for the boundary layer equations, with and without mass transfer, and for the incompressible viscous flow in a driven cavity. Divergence and nondivergence equations are considered for the cavity. Among the fourth-order techniques, it is shown that spline 4 has the smallest truncation error. The spline 4 procedure generally requires one-quarter the number of mesh points in a given coordinate direction as a central finite-difference calculation of equal accuracy. The Hermite 6 procedure leads to remarkably accurate boundary layer solutions.

Rubin, S. G.↗